Coarse Proximity and Proximity at Infinity

TL;DR

This paper introduces coarse proximity structures as a large-scale analog of small-scale proximity spaces, constructs a natural proximity at infinity for unbounded metric spaces, and shows it is a coarse invariant.

Contribution

It defines coarse proximity structures, constructs the proximity at infinity, and proves its functoriality and invariance under coarse isomorphisms.

Findings

Coarse proximity structures generalize small-scale proximity to large-scale spaces.

The proximity at infinity is functorial and a coarse invariant.

The construction applies to unbounded metric spaces, capturing their large-scale geometry.

Abstract

We define coarse proximity structures, which are an analog of small-scale proximity spaces in the large-scale context. We show that metric spaces induce coarse proximity structures, and we construct a natural small-scale proximity structure, called the proximity at infinity, on the set of equivalence classes of unbounded subsets of an unbounded metric space given by the relation of having finite Hausdorff distance. We show that this construction is functorial. Consequently, the proximity isomorphism type of the proximity at infinity of an unbounded metric space $X$ is a coarse invariant of $X$.